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UNIT-II : Measures of Central Tendency and Dispersion: Meaning and Objectives of Measures of Central Tendency, Different Measure Viz

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UNIT-II : Measures of Central Tendency and Dispersion: Meaning and Objectives of Measures of Central Tendency, Different Measure Viz 1 BUSINESS STATISTICS BBA UNIT –II, 2ND SEM UNIT-II : Measures of Central Tendency and Dispersion: Meaning and objectives of measures of central tendency, different measure viz. arithmetic mean, median, mode, geometric mean and harmonic mean, characteristics, applications and limitations of these measures; measure of variation viz. range, quartile deviation mean deviation and standard deviation, co-efficient of variation and skewness. Measures of Central Tendency: A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistic: 1. Mean 2. Median 3. Mode The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used. 1. Mean: There are 3 types of mean 1A. Arithmetic Mean The mean (or average) is the most popular and well known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous data. The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if we have n values in a data set and they have values x1, x2, ..., xn, the sample mean, usually denoted by (pronounced x bar), is: This formula is usually written in a slightly different manner using the Greek capitol letter, , pronounced "sigma", which means "sum of...": Example: The marks of seven students in a mathematics test with a maximum possible mark of 20 are given below: 15 13 18 16 14 17 12 Dr. Kajarii Roy, Professor, MBA Department, Somany Institute of Technology and Management, Haryana. 2 BUSINESS STATISTICS BBA UNIT –II, 2ND SEM Find the mean of this set of data values. Solution: So, the mean mark is 15 Advantages : 1. It can be easily calculated; and can be easily understood. It is the reason that it is the most used measure of central tendency. 2. As every item is taken in calculation, it is effected by every item. 3. As the mathematical formula is rigid one, therefore the result remains the same. 4. Fluctuations are minimum for this measure of central tendency when repeated samples are taken from one and the same population. 5. It can further be subjected to algebraic treatment unlike other measures i.e. mode and median. 6. A.M. has also a plus point being a calculated quantity and is not based on position of terms in a series. 7. As it is rigidly defined, it is mostly used for comparing the various issues. Disadvantages: 1. The arithmetic mean is highly affected by extreme values. 2. It cannot average the ratios and percentages properly. 3. It is not an appropriate average for highly skewed distributions. 4. It cannot be computed accurately if any item is missing. 5. The mean sometimes does not coincide with any of the observed value. 1B: Geometric Mean A geometric mean is a mean or average which shows the central tendency of a set of numbers by using the product of their values. For a set of n observations, a geometric mean is the nth root of their product. The geometric mean G.M., for a set of numbers x1, x2, … , xn is given as 1⁄n G.M. = (x1. x2 … xn) n 1⁄n n or, G. M. = (π i = 1 xi) = √( x1, x2, … , xn). Dr. Kajarii Roy, Professor, MBA Department, Somany Institute of Technology and Management, Haryana. 3 BUSINESS STATISTICS BBA UNIT –II, 2ND SEM The geometric mean of two numbers, say x, and y is the square root of their product x×y. For three numbers, it will be the cube root of their products i.e., (x y z) 1⁄3. Geometric Mean of Frequency Distribution For a grouped frequency distribution, the geometric mean G.M. is f1 f2 fn 1⁄N n G.M. = (x1 . x2 … xn ) , where N = ∑ i= 1 fi Taking logarithms on both sides, we get n log G.M. = 1⁄N (f1 log x1 + f2 log x2 + … + fn log xn) = 1⁄N [∑ i= 1 fi log xi ]. Properties of Geometric Mean The logarithm of geometric mean is the arithmetic mean of the logarithms of given values If all the observations assumed by a variable are constants, say K >0, then the G.M. of the observation is also K The geometric mean of the ratio of two variables is the ratio of the geometric means of the two variables The geometric mean of the product of two variables is the product of their geometric means Geometric Mean of a Combined Group Suppose G1, and G2 are the geometric means of two series of sizes n1, and n2 respectively. The geometric mean G, of the combined groups, is: log G = (n1 log G1 + n2 log G2) ⁄ (n1 + n2) or, G = antilog [(log G1 + n2 log G2) ⁄ (n1 + n2)] In general for ni geometric means, i = 1 to k, we have G = antilog [(log G1 + n2 log G2 + … + nk log Gk) ⁄ (n1 + n2 + … +nk)] Advantages: A geometric mean is based upon all the observations It is rigidly defined The fluctuations of the observations do not affect the geometric mean It gives more weight to small items Dr. Kajarii Roy, Professor, MBA Department, Somany Institute of Technology and Management, Haryana. 4 BUSINESS STATISTICS BBA UNIT –II, 2ND SEM Disadvantages : A geometric mean is not easily understandable by a non-mathematical person If any of the observations is zero, the geometric mean becomes zero If any of the observation is negative, the geometric mean becomes imaginary 1C: Harmonic Mean Harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the observations. The most important criteria for it is that none of the observations should be zero. A harmonic mean is used in averaging of ratios. The most common examples of ratios are that of speed and time, cost and unit of material, work and time etc. The harmonic mean (H.M.) of n observations is n H.M. = 1÷ (1⁄n ∑ i= 1 (1⁄xi) ) In the case of frequency distribution, a harmonic mean is given by n n H.M. = 1÷ [1⁄N (∑ i= 1 (fi ⁄ xi)], where N = ∑ i= 1 fi Properties of Harmonic Mean If all the observation taken by a variable are constants, say k, then the harmonic mean of the observations is also k The harmonic mean has the least value when compared to the geometric mean and the arithmetic mean Advantages: A harmonic mean is rigidly defined It is based upon all the observations The fluctuations of the observations do not affect the harmonic mean More weight is given to smaller items Disadvantages: Not easily understandable Difficult to compute Numericals: Dr. Kajarii Roy, Professor, MBA Department, Somany Institute of Technology and Management, Haryana. 5 BUSINESS STATISTICS BBA UNIT –II, 2ND SEM 1. Suppose a person cover a certain distance d at a speed of x. He returns back to the starting point with a speed of y. In this case, the average speed of the person is calculated by the harmonic mean. Average speed = Total distance covered / Total time taken = 2d (d⁄x + d⁄y). In other words, if an equal distance is covered with speeds S1, S2, … , Sn, then Average speed = n ÷ ∑ (1⁄S). If different distances D1, D2, … , Dn, is covered with different speeds S1, S2, … , Sn, the average speed is n n Average Speed = [∑ i= 1 Di] ⁄ [∑ i= 1 (Di ⁄ Si)] 2. Calculate the geometric and harmonic mean of the given data x 2 4 5 8 f 3 3 2 2 f1 f2 fn 1⁄N Solution: Geometric mean = G.M. = (x1 . x2 … xn ) Here, N = 3 + 3 + 2 + 2 = 10 G.M. = (2 3 × 4 3 × 5 2 × 8 2)1/10 or, G.M. =(8×64×25×64)1/10 = (819200)1/10 n H.M. = 1 ÷ [1 ⁄ N ∑ i= 1 (fi ⁄ xi) ] = 1÷[1⁄10 × (3⁄2 + ¾ + 2⁄5 + 2⁄8)] = 100⁄29. If AM = arithmetic mean, GM = geometric mean, and HM = harmonic mean. The relationship between the three is given by the formula : AM x HM = GM2 Let there are two numbers ‘a’ and ‘b’, a, b > 0 then AM = a+b/2 GM =√ab HM =2ab/a+b ∴ AM × HM =a+b/2 × 2ab/a+b = ab = (√ab)2 = (GM)2 Note that these means are in G.P. Hence AM.GM.HM follows the rules of G.P. Dr. Kajarii Roy, Professor, MBA Department, Somany Institute of Technology and Management, Haryana. 6 BUSINESS STATISTICS BBA UNIT –II, 2ND SEM i.e. G.M. =√A.M. × H.M. Now, let us see the difference between AM and GM AM – GM =a+b/2 – √ab =(√a2)+(√b)–2√a√b/2 i.e. AM > GM Similarly, G.M. – H.M. = √ab –2ab/a+b =√ab/a+b (√a – √b)2 > 0 So. GM > HM 2. Median The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value. Example: The marks of nine students in a geography test that had a maximum possible mark of 50 are given below: 47 35 37 32 38 39 36 34 35 Find the median of this set of data values.
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